# Probability of success. Formula: n = number of trials k = number of successes n – k = number of failures p = probability of success in one trial q = 1 – p = probability of failure in one trial. Example: You are taking a 10 question multiple choice test. If each question has four choices and you guess on each question, what is the probability of getting.

## Probability of success. The probability that neither trial is successful is (15/16)2 (assuming that the trials are independent), and the chances that at least one trial is successful is one minus that: 1−(15/16)2=31/

Predictive probability of success PPOS is a statistics concept commonly used in the pharmaceutical industry including by health authorities to support decision making. In clinical trials , PPOS is the probability of observing a success in the future based on existing data. It is one type of probability of success.

A Bayesian means by which the PPOS can be determined is through integrating the data's likelihood over possible future responses posterior distribution. Conditional power is the probability of observing a statistically significance assuming the parameter equals to a specific value. Conditional power is often criticized for assuming the parameter equals to a specific value which is not known to be true. If the true value of the parameter is known, there is no need to do an experiment.

Predictive power addresses this issue assuming the parameter has a specific distribution. Predictive power is a Bayesian power. A parameter in Bayesian setting is a random variable. Predictive power is a function of a parameter s , therefore predictive power is also a variable. Both conditional power and predictive power use statistical significance as success criteria. However statistical significance is often not enough to define success.

For example, health authorities often require the magnitude of treatment effect to be bigger than statistical significance to support a registration decision. To address this issue, predictive power can be extended to the concept of PPOS.

The success criteria for PPOS is not restricted to statistical significance. It can be something else such as clinical meaningful results. PPOS is conditional probability conditioned on a random variable, therefore it is also a random variable. The observed value is just a realization of the random variable. Posterior probability of success is calculated from posterior distribution. PPOS is calculated from predictive distribution. Posterior distribution is the summary of uncertainties about the parameter.

Predictive distribution has not only the uncertainty about parameter but also the uncertainty about estimating parameter using data. Posterior distribution and predictive distribution have same mean, but former has smaller variance.

PPOS is a conditional probability conditioned on randomly observed data and hence is a random variable itself. Currently common practice of PPOS uses only its point estimate in applications.

This can be misleading. For a variable, the amount of uncertainty is an important part of the story. To address this issue, Tang [5] introduced PPOS credible interval to quantify the amount of its uncertainty. Tang advocates to use both PPOS point estimate and credible interval in applications such as decision making and clinical trial designs.

Another common issue is the mixed use of posterior probability of success and PPOS. As described in the previous section, the 2 statistics are measured in 2 different metrics, comparing them is like comparing apples and oranges.

Traditional pilot trial design is typically done by controlling type I error rate and power for detecting a specific parameter value. The goal of a pilot trials such as a phase II trial is usually not to support registration. Therefore, it doesn't make sense to control type I error rate especially a big type I error as typically done in a phase II trial. Therefore, it makes more sense to design a trial based on PPOS. However the PPOS can be small just due to chance. Finding an optimal design is equivalent to find the solution to the following 2 equations.

The first equation ensures that the PPOS is small such that not too many trials will be prevented entering next stage to guard against false negative.

The first equation also ensures that the PPOS is not too small such that not too many trials will enter the next stage to guard against false positive. The second equation also ensures that the PPOS credible interval is not too tight such that it won't demand too much resource. PPOS can also be used in Interim analysis to determine whether a clinical trial should be continued.

PPOS can be used for this purpose because its value can be use to indicate if there is enough convincing evidence to either reject or fail to reject the null hypothesis with the presently available data. Traditional futility interim is designed based on beta spending. However beta spending doesn't have intuitive interpretation.

Therefore, it is difficult to communicate with non-statistician colleagues. Finding the optimal design is equivalent to solving the following 2 equations. In interim analysis, Predictive Probability of Success can also be calculated through the use of simulations through the following method: Using simulation to calculate PPOS makes it possible to test statistics with complex distributions since it alleviates the computing complexity that would otherwise be required.

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